Atoms and Nuclei

Black Body Radiation

Query 8:Do not all fix’d
Bodies, when heated beyond a certain degree, emit Light and shine; and is not
this Emission perform’d by the vibrating motion of its parts?

Isaac Newton, Opticks, published 1704.

Heated Bodies Radiate

We shall now turn to another puzzle
confronting physicists at the turn of the century (1900): just how do heated
bodies radiate? There was a general
understanding of the mechanism involved—heat was known to cause the molecules
and atoms of a solid to vibrate, and the molecules and atoms were themselves
complicated patterns of electrical charges. (As usual, Newton was on the right track.) From the
experiments of Hertz and others, Maxwell’s predictions that oscillating charges
emitted electromagnetic radiation had been confirmed, at least for simple
antennas. It was known from Maxwell’s
equations that this radiation traveled at the speed of light and from this it
was realized that light itself, and the closely related infrared heat
radiation, were actually electromagnetic waves. The picture, then, was that when a body was heated, the consequent
vibrations on a molecular and atomic scale inevitably induced charge oscillations. Assuming then that Maxwell’s theory of
electromagnetic radiation, which worked so well in the macroscopic world, was
also valid at the molecular level, these oscillating charges would radiate,
presumably giving off the heat and light observed.

How is Radiation Absorbed?

What is meant by the phrase “black
body” radiation? The point is that the
radiation from a heated body depends to some extent on the body being
heated. To see this most easily, let’s
back up momentarily and consider how different materials absorb radiation. Some, like glass, seem to
absorb light hardly at all—the light goes right through. For a shiny metallic surface, the light isn’t
absorbed either, it gets reflected. For
a black material like soot, light and heat are almost completely absorbed, and
the material gets warm. How can we
understand these different behaviors in terms of light as an electromagnetic
wave interacting with charges in the material, causing these charges to
oscillate and absorb energy from the radiation? In the case of glass, evidently this doesn’t happen, at least not
much. Why not? A full understanding of why needs quantum
mechanics, but the general idea is as follows: there are charges—electrons—in glass that are able to oscillate in
response to an applied external oscillating electric field, but these charges are tightly bound to
atoms, and can only oscillate at certain frequencies. (For quantum experts, these charge
oscillations take place as an electron moves from one orbit to another. Of course, that was not understood in the
1890’s, the time of the first precision work on black body radiation.) It happens that for ordinary glass none of these frequencies corresponds to
visible light, so there is no resonance with a light wave, and hence little energy absorbed. That’s why glass is perfect for windows! Duh. But glass is opaque at some
frequencies outside the visible range
(in general, both in the infrared and the ultraviolet). These are the frequencies at which the
electrical charge distributions in the atoms or bonds can naturally oscillate.

How can we understand the reflection of light by a metal surface? A piece of metal has electrons free to move
through the entire solid. This is what
makes a metal a metal: it conducts both electricity and heat easily, both are
actually carried by currents of these freely moving electrons. (Well, a little of the heat is carried by
vibrations.) But metals are recognizable
because they’re shiny—why’s that? Again, it’s those free electrons: they’re driven into large (relative to
the atoms) oscillations by the electrical field of the incoming light wave, and
this induced oscillating current radiates electromagnetically, just like a
current in a transmitting antenna. This
radiation is the reflected
light. For a shiny metal surface, little
of the incoming radiant energy is absorbed as heat, it’s just reradiated, that
is, reflected.

Now let’s consider a substance that absorbs light: no transmission and no
reflection. We come very close to perfect absorption with soot. Like a metal, it will conduct an electric
current, but nowhere near as efficiently. There are unattached
electrons, which can move through the whole solid, but they constantly bump
into things—they have a short mean free path. When they bump, they cause vibration, like balls hitting bumpers in a
pinball machine, so they give up kinetic energy into heat. Although the electrons in soot have a short
mean free path compared to those in a good metal, they move very freely
compared with electrons bound to atoms (as in glass), so they can accelerate
and pick up energy from the electric field in the light wave. They are therefore very effective
intermediaries in transferring energy from the light wave into heat.

Relating Absorption and Emission

Having seen how soot can absorb
radiation and transfer the energy into heat, what about the reverse? Why does
it radiate when heated? The pinball machine analogy is still good: imagine now
a pinball machine where the barriers, etc., vibrate vigorously because they are
being fed energy. The balls (the electrons) bouncing off them will be suddenly
accelerated at each collision, and these accelerating charges emit
electromagnetic waves. On the other
hand, the electrons in a metal have
very long mean free paths, the lattice vibrations affect them much less, so
they are less effective in gathering and radiating away heat energy. It is evident from considerations like this
that good absorbers of radiation are also good emitters.

In fact, we can be much more precise: a body emits radiation at a given temperature
and frequency exactly as well as it
absorbs the same radiation. This was
proved by Kirchhoff: the essential point is that if we suppose a particular
body can absorb better than it emits, then in a room full of objects all at the
same temperature, it will absorb radiation from the other bodies better than it
radiates energy back to them. This means
it will get hotter, and the rest of the room will grow colder, contradicting
the second law of thermodynamics. (We
could use such a body to construct a heat engine extracting work as the room
grows colder and colder!)

But a metal glows when it’s heated
up enough: why is that? As the temperature is raised, the lattice of atoms
vibrates more and more, these vibrations scatter and accelerate the electrons. Even glass glows at high enough temperatures,
as the electrons are loosened and vibrate.

The “Black Body” Spectrum: a Hole in the Oven

Any body at any temperature above
absolute zero will radiate to some extent, the intensity and frequency
distribution of the radiation depending on the detailed structure of the
body. To begin analyzing heat radiation,
we need to be specific about the body doing the radiating: the
simplest possible case is an idealized body which is a perfect absorber, and
therefore also (from the above argument) a perfect emitter. For obvious
reasons, this is called a “black body”.

But we need to check our ideas
experimentally: so how do we construct a perfect absorber? OK, nothing’s perfect, but in 1859 Kirchhoff
had a good idea: a small hole in the side of a large box is an excellent
absorber, since any radiation that goes through the hole bounces around inside,
a lot getting absorbed on each bounce, and has little chance of ever getting
out again. So, we can do this in reverse: have an oven with a tiny
hole in the side, and presumably the radiation coming out the hole is as good a
representation of a perfect emitter as we’re going to find. Kirchhoff challenged theorists and
experimentalists to figure out and measure (respectively) the energy/frequency
curve for this “cavity radiation”, as he called it (in German, of course:
hohlraumstrahlung, where hohlraum means hollow room or cavity, strahlung is
radiation). In fact, it was Kirchhoff’s challenge in 1859 that led directly to
quantum theory forty years later!

What Was Observed: Two Laws

The first quantitative conjecture
based on experimental observation of hole radiation was:

Stefan’s
Law (1879): the total power P radiated from one square meter of black
surface at temperature T goes as the fourth
power of the absolute temperature:

Five years later, in 1884, Boltzmann
derived this T 4 behavior
from theory: he applied classical thermodynamic reasoning to a box filled with
electromagnetic radiation, using Maxwell’s equations to relate pressure to
energy density. (The tiny amount of energy coming out of the hole would of
course have the same temperature dependence as the radiation intensity
inside.) See the accompanying notes for
details of the derivation.

Exercise: the sun’s surface temperature is 5700K. How much power is radiated by one square
meter of the sun’s surface? Given that
the distance to earth is about 200 sun radii, what is the maximum power
possible from a one square kilometer solar energy installation?

Another important finding was Wien’s Displacement Law:

As the oven temperature varies, so
does the frequency at which the emitted radiation is most intense. In fact,
that frequency is directly proportional to the absolute temperature:

(Wien himself deduced this law
theoretically in 1893, following Boltzmann’s thermodynamic reasoning. It had
previously been observed, at least semi-quantitatively, by an American
astronomer, Langley.) The formula is derived in the accompanying notes.

In fact, this upward shift in fmax with T is familiar to everyone—when an iron is heated in a fire, the
first visible radiation (at around 900K) is deep red, the lowest frequency
visible light. Further increase in T causes the color to change to orange then yellow, and finally blue at very high
temperatures (10,000K or more) for which the peak in radiation intensity has
moved beyond the visible into the ultraviolet.

This shift in the frequency at which
radiant power is a maximum is very important for harnessing solar energy, such
as in a greenhouse. The glass must allow
the solar radiation in, but not let the heat radiation out. This is feasible because the two radiations
are in very different frequency ranges—5700K and, say, 300K—and there are
materials transparent to light but opaque to infrared radiation. Greenhouses only work because fmax varies with temperature.

What Was Observed: the Complete Picture

By the 1890’s, experimental
techniques had improved sufficiently that it was possible to make fairly
precise measurements of the energy distribution in this cavity radiation, or as
we shall call it black body radiation. In 1895, at the
University
of
Berlin, Wien and Lummer
punched a small hole in the side of an otherwise completely closed oven, and
began to measure the radiation coming out.

The beam coming out of the hole was
passed through a diffraction grating, which sent the different wavelengths/frequencies
in different directions, all towards a screen. A detector was moved up and down along the screen to find how much
radiant energy was being emitted in each frequency range. (This is a theorist’s
model of the experiment—actual experimental arrangements were much more
sophisticated. For example, to make the
difficult infrared measurements higher frequency waves were eliminated by
multiple reflections from quartz and other crystals.) They found a radiation intensity/frequency
curve close to this (correct one):

The visible spectrum begins at
around 4.3×1014 Hz, so this oven glows deep red.

One
minor point: thisplot is the energy density inside the oven, which we denote by ρ(f,
T), meaning that at temperature T,
the energy in Joules/m3 in the frequency interval f, f + Δf is ρ(f, T)Δf.

To find the power pumped out of the
hole, bear in mind that the radiation inside the oven has waves equally going
both ways—so only half of them will come out through the hole. Also, if the hole has area A, waves coming from the inside at an
angle will see a smaller target area. The result of these two effects is that the

They were also able to confirm both
Stefan’s Law P = σT4 and Wien’s Displacement Law by measuring the
black body curves at different temperatures, for example:

Let’s look at these curves in more
detail: for low frequencies f, ρ( f, T) was found to be proportional to f 2, a parabolic shape, but for increasing f it fell below the parabola, peaking at fmax,
then dropping quite rapidly towards zero as f increased beyond fmax.

For those low frequencies where ρ( f, T) is parabolic, doubling the temperature was
found to double the intensity of the radiation. But also at 2T the curve followed the doubled parabolic path much further before dropping away—in fact,
twice as far, and fmax(2T)
= 2fmax(T).

The curve ρ( f, 2T), then, reaches eight times the height of ρ( f, T). (See the graph above.) It also spreads over twice the lateral extent,
so the area under the curve, corresponding to the total energy radiated,
increases sixteenfold on doubling the temperature: Stefan’s Law, P = σT4.

Understanding the Black Body Curve

These beautifully precise
experimental results were the key to a revolution. The first successful theoretical analysis of
the data was by Max Planck in 1900. He
concentrated on modeling the oscillating charges that must exist in the oven walls,
radiating heat inwards and—in thermodynamic equilibrium—themselves being driven
by the radiation field.

The bottom line is that he found he
could account for the observed curve if he required these oscillators not to radiate
energy continuously, as the classical theory would demand, but they could only lose or gain energy in
chunks, called quanta, of size hf, for an oscillator of frequency f. The constant h is now called
Planck’s constant, h = 6.626 × 10-34 joule.sec.

With that assumption, Planck
calculated the following formula for the radiation energy density inside the
oven:

The perfect agreement of this
formula with precise experiments, and the consequent necessity of energy
quantization, was the most important
advance in physics in the century.

But no-one noticed for several
years! His black body curve was
completely accepted as the correct one: more and more accurate experiments
confirmed it time and again, yet the radical nature of the quantum assumption
didn’t sink in. Planck wasn’t too
upset—he didn’t believe it either, he saw it as a technical fix that (he hoped)
would eventually prove unnecessary.

Part of the problem was that Planck’s route to the formula was
long, difficult and implausible—he even made contradictory assumptions at
different stages, as Einstein pointed out later. But the result was correct anyway, and to
understand why we’ll follow another, easier, route initiated (but not
successfully completed) by Lord Rayleigh in
England
.

Rayleigh’s Sound Idea: Counting Standing Waves

In 1900, actually some months before
Planck’s breakthrough work, Lord Rayleigh was taking a more direct approach to
the radiation inside the oven: he didn’t even think about oscillators in the
walls, he justtook the radiation to be a collection of standing waves in a cubical
enclosure: electromagnetic oscillators. In contrast to the somewhat murky reality of the wall oscillators, these
standing electromagnetic waves were crystal clear.

This was a natural approach for Rayleigh—he’d
solved an almost identical problem a quarter century earlier, an analysis of
standing sound waves in a cubical
room (§267 of his book). The task is to
find and enumerate the different possible standing waves in the room/oven,
compatible with the boundary conditions. For sound waves in a room, the amplitude of the sound goes to zero at
the walls. For the electromagnetic waves, the electric field parallel to the
wall must go to zero if the wall is a perfect conductor (and it’s OK to assume
this—see note later).

So what are the allowed standing
waves? As a warm up exercise, consider
the different allowed modes of vibration, that is, standing waves, in a string
of length a fixed at both ends:

The possible values of wavelength
are:

So the allowed frequencies are

These allowed frequencies are
equally spaced c/2a apart. We define the spectral density by stating that

number
of modes between f and f + Δf = N(f )Δf

where we assume that Δf is large compared with the spacing
between successive frequencies. Evidently for this one-dimensional exercise N(f ) is a constant equal to 2a/c, each mode corresponds to an integer
point on the real axis in units c/2a.

The amplitude of oscillation as a
function of time is:

more conveniently written

The allowed values of k (called the wave number) are:

The generalization to three
dimensions is simple: in a cubical box of side a, an allowed standing wave must satisfy the boundary conditions in
all three directions. This means the
choices of wave numbers are:

That is to say, each modes is
labeled with three positive integers:

and the frequency of the mode is:

(Details of the electromagnetic
waves, and derivation of this formula, are given in the accompanying notes.)

For infrared and visible radiation in
a reasonable sized oven, frequency intervals measured experimentally are far
greater than the spacing c/2a of these integer points. Just as in the one-dimensional example, these
modes fill the three-dimensional k-space
uniformly, with density (a/π)3,
but now this means the mode density is not uniform as a function of frequency.

The number of them between f and f + Δf = N(f )Δf is the volume in k-space, in units (π/a)3, of the spherical shell
of radius k = 2πf/c,
thickness Δk = 2πΔf/c,
and restricted to all components of k being positive (like the integers), a factor of 1/8.

Including a factor of 2 for the two
polarization states of the standing electromagnetic waves, the density of
states as a function of frequency in an oven of volume V = a 3 is:

giving the density of radiation
states in the oven

(Details of this
analysis can be found in the notes. If
you’re wondering why it’s OK to have an oven with essentially perfectly
reflecting walls when we were previously insisting on absorbing walls,
Kirchhoff proved long before that two such ovens at the same temperature will
have the same radiation intensity—otherwise energy could be transferred from
one to the other, violating the Second Law.)

What about Equipartition of Energy?

A central result of classical
statistical mechanics is the equipartition of energy: for a system in thermal
equilibrium, each degree of freedom has average energy ½kT. Thus molecules in a gas
have average kinetic energy 3/2kT, ½kT for each direction, and a simple
one-dimensional harmonic oscillator has total energy kT: ½kT kinetic energy and ½kT potential energy.

Comparing now the formula for the
number of modes N(f )Δf in a small interval Δf

with Planck’s formula for radiation
energy intensity in the same interval:

For the low frequency modes hf << kT we can make the approximation

and it follows immediately that each mode has energykT.

But things go badly wrong at high
frequencies! The number of modes
increases without limit, the energy in these high frequency modes, though, is decaying exponentially as the frequency increases. Ehrenfest later dubbed this the ultraviolet catastrophe. Rayleigh’s sound approach apparently wasn’t
so sound after all—something crucial was missing.

It is perhaps surprising that Planck never mentioned equipartition. Of course, as Rayleigh himself remarked,
equipartition was well-known to have problems, for example in the specific heat
of gases. And in fact Planck wasn’t even sure about the existence of atoms: he later
wrote that in the 1890’s “I had been inclined to reject atomism” (see
notes). In fact, even Boltzmann was very
unsure how well oscillators came to thermal equilibrium with electromagnetic
radiation—after all, it was well known that oscillation of diatomic molecules
failed to reach classical thermal equilibrium with kinetic energy. (As long ago as 1877, Maxwell had pointed out
that hot gases emit light at particular frequencies. The frequencies do not change with
temperature, so the oscillations must be simple harmonic—but such an oscillator
would surely also be excited by collisions at low temperatures, so why was energy not being fed into this mode?)

Einstein Sees a Gas of Photons

As mentioned earlier, after Planck
announced his result in December, 1900 there was a deafening silence on the
subject for several years. No-one
(including Planck) realized the importance of what he had done—his work was
widely seen as just a clever technical fix, even if it did give the right
answer (the curve itself was completely accepted as correct).

Then in March, 1905, Albert Einstein
turned his attention to the problem. He
first rederived the Rayleigh result assuming equipartition:

and observed that this made no
sense at high frequencies. So he focused
on Planck’s formula for high frequencies, hf >> kT:

(actually identical in this region to an
earlier formula by Wien).

Einstein perceived an analogy here
with the energy distribution in a classical gas.

Recall from the last lecture that
the (normalized) probability distribution function for classical atoms as a
function of speed v was

,

and the corresponding energy
density in v is

The radiation formula at high
frequencies is

Einstein pointed out that if the high
frequency radiation is imagined to be a gas of independent particles having
energy E = hf, the energy density in frequency in the radiation is

Comparing this with the expression
for atoms, the analogy is close: recall that for the radiation, frequency is
proportional to wave number and, on quantization, to momentum; for the
(nonrelativistic) atoms velocity is proportional to momentum, so both these
distributions are essentially in momentum space. Of course, the normalization factors differ,
because the total number of atoms doesn’t change with temperature, unlike the
total radiation. Nevertheless, the
analogy is compelling, and led Einstein to state that the radiation in the enclosure was itself quantized, the energy
quantization was not some special
property only of the wall oscillators, as Planck thought. The radiation quanta are of course photons,
but that word wasn’t coined until later.

Einstein had been troubled by Planck’s
derivation of his result, depending as it did first, on a classical analysis of
the interaction between the wall oscillator and radiation, followed by a claim
that the interaction was in fact not like that at all. But the answer was right, and now Einstein
began to see why. In contrast to the
poorly understood wall oscillators, the electromagnetic standing wave
oscillations in the oven were completely clear.

Energy in an Oscillator as a Function of Temperature

Einstein realized that, in terms of
Rayleigh’s electromagnetic standing waves, the blackbody radiation curves have
a simple interpretation: the average energy in an oscillator of frequency f at temperature T is

Furthermore, Planck’s work made
plausible that this same quantization held for the material oscillators in the
walls.

Einstein took the next step: he
conjectured that all oscillators are quantized, for example a vibrating atom in
a solid. This would explain why the
Dulong Petit law, which assigns specific heat 3k to each atom in a solid, does not hold good at low temperatures:
once kT << hf, the modes are not excited, so absorb little
heat. The specific heat falls, as is
indeed observed. Furthermore, it
explains why diatomic gas molecules, such as oxygen and nitrogen, do not appear
to absorb heat into vibrational modes—these modes have very high frequency.

It’s worth thinking about the constant exchange of energy
with the environment for an oscillator in thermal equilibrium at temperature T. The random thermal fluctuations in a system have energy of order kT, this is the amount of energy,
approximately, delivered back and forth. But if an oscillator has hf =
5kT, say, it can only accept chunks
of energy of size 5kT, and will only
be excited in the unlikely event that five of these random kT fluctuations come together at the right place at the right
time. The high frequency modes are
effectively frozen out by this minimum energy requirement. The exponential drop off in excitation with
frequency reflects the exponential drop off in probability of getting the right
number of fluctuations together, analogous to the exponential drop off in
probability of tossing a coin n heads
in a row.

Simple Derivation of Planck’s Formula from the Boltzmann’s Distribution

Planck’s essential assumption in
deriving his formula was that the oscillators only exchange energy with the
radiation in quanta hf. Einstein made clear that the well-understood
standing electromagnetic waves, the radiation in the oven, also have quantized
energies.

As discussed in the previous
lecture, the probability of a system at temperature T having energy E is
proportional to Boltzmann’s
formula. It turns out that this formula
continues to be valid in quantum systems. Now, a classical simple harmonic oscillator at T will have a probability distribution proportional to so the expectation value of the energy is

just the classical equipartition of
energy.

But we now know this isn’t true if
the oscillator is quantized: the
energies are now in steps hf apart. Taking the ground state as the
zero of energy, allowed energies are

0, hf, 2hf,
3hf, …

and assuming the Boltzmann
expression for relative probabilities is still correct, the relative
probabilities of these states will be in the ratios:

To find the oscillator energy at
this temperature, we use these probabilities weighted by the corresponding
energy, and divide by a normalization factor to ensure that the probabilities
add up to 1:

This is indeed the correct result
from the black body experiments. Evidently Boltzmann’s relative probability
function is still valid in quantum systems.

A Note on Wien’s Displacement Law

It is easy to see how Wien’s
Displacement Law follows from Planck’s formula: the maximum radiation per unit
frequency range is at the frequency f for which the function is a maximum. Solving
numerically gives hfmax=2.82kT.

It can be established theoretically
(and is confirmed experimentally) that the equation connecting the frequency of
maximum energy intensity in units of Joules/m3/Hz is:

However, the law is often stated in
terms of the wavelength at which the
intensity, now measured in Joules/m3/m, that is, per unit interval
of wavelength, and

The important point to notice here
is that these formulas do not give the same result, as is easily verified,
since , not the speed of light! The reason is that the two measures, per unit interval of frequency and
per unit interval of wavelength, are different, so a claim that, say, sunlight
is most intense in the yellow has to specify which is being used (actually it
would be wavelength, frequency would give the near infrared).

The graphs of black body radiation
as a function of temperature were generated using an Excel spreadsheet. You are
welcome to download this spreadsheet and use it to explore how radiation varies
with temperature. It’s very easy to use—you just put in the temperature and
watch the graph change.

A nice example of black body
radiation is that left over from the Big Bang. It has been found that the
intensity pattern of this background radiation in the Universe follows the
black body curve very precisely, for a temperature of about three degrees above
absolute zero. More details can be found here.